CONTENTS 1. Introduction...............................................................................5 2. Spaces of measurable functions...............................................7 3. Proper domain of an integral transformation...........................14 4. Integral transformations in L⁰. Continuity and closibility...........17 5. Extensions by continuity. Compatibility problem.......................21 6. Compactness of integral transformations................................35 7. Miscellaneous results and comments......................................41 8. Bibliographical remarks and comments...................................46 Bibliography.................................................................................48
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We have shown in [1] that domains of integral operators are not in general locally convex. In the case when such a domain is locally convex we show that it is an inductive limit of L¹-spaces with weights.
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It is shown that the proper domains of integral operators have separating duals but in general they are not locally convex. Banach function spaces which can occur as proper domains are characterized. Some known and some new results are given, illustrating the usefulness of the notion of proper domain.
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We prove the $L^p$ boundedness of certain nonconvolutional oscillatory integral operators and give explicit description of their extended domains. The class of phase functions considered here includes the function $|x|^{α}|y|^{β}$. Sharp boundedness results are obtained in terms of α, β, and rate of decay of the kernel at infinity.
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